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{
"cells": [
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Code for Finding a large prime number (min 20-digit) "
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": [
"from random import randrange, getrandbits\n",
"def is_prime(n, k=128):\n",
" \"\"\" Test if a number is prime\n",
" Args:\n",
" n -- int -- the number to test\n",
" k -- int -- the number of tests to do\n",
" return True if n is prime\n",
" \"\"\"\n",
" # Test if n is not even.\n",
" # But care, 2 is prime !\n",
" if n == 2 or n == 3:\n",
" return True\n",
" if n <= 1 or n % 2 == 0:\n",
" return False\n",
" # find r and s\n",
" s = 0\n",
" r = n - 1\n",
" while r & 1 == 0:\n",
" s += 1\n",
" r //= 2\n",
" # do k tests\n",
" for _ in range(k):\n",
" a = randrange(2, n - 1)\n",
" x = pow(a, r, n)\n",
" if x != 1 and x != n - 1:\n",
" j = 1\n",
" while j < s and x != n - 1:\n",
" x = pow(x, 2, n)\n",
" if x == 1:\n",
" return False\n",
" j += 1\n",
" if x != n - 1:\n",
" return False\n",
" return True\n",
"def generate_prime_candidate(length):\n",
" \"\"\" Generate an odd integer randomly\n",
" Args:\n",
" length -- int -- the length of the number to generate, in bits\n",
" return a integer\n",
" \"\"\"\n",
" # generate random bits\n",
" p = getrandbits(length)\n",
" # apply a mask to set MSB and LSB to 1\n",
" p |= (1 << length - 1) | 1\n",
" return p\n",
"def generate_prime_number(length=60):\n",
" \"\"\" Generate a prime\n",
" Args:\n",
" length -- int -- length of the prime to generate, in bits\n",
" return a prime\n",
" \"\"\"\n",
" p = 4\n",
" # keep generating while the primality test fail\n",
" while not is_prime(p, 128):\n",
" p = generate_prime_candidate(length)\n",
" return p\n",
"print(generate_prime_number())"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"610292520201647"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Code for finding Primitive root "
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": [
"# Python3 program to find primitive root \n",
"# of a given number n \n",
"from math import sqrt \n",
" \n",
"# Returns True if n is prime \n",
"def isPrime( n): \n",
" \n",
" # Corner cases \n",
" if (n <= 1): \n",
" return False\n",
" if (n <= 3): \n",
" return True\n",
" \n",
" # This is checked so that we can skip \n",
" # middle five numbers in below loop \n",
" if (n % 2 == 0 or n % 3 == 0): \n",
" return False\n",
" i = 5\n",
" while(i * i <= n): \n",
" if (n % i == 0 or n % (i + 2) == 0) : \n",
" return False\n",
" i = i + 6\n",
" \n",
" return True\n",
" \n",
"\"\"\" Iterative Function to calculate (x^n)%p \n",
" in O(logy) */\"\"\"\n",
"def power( x, y, p): \n",
" \n",
" res = 1 # Initialize result \n",
" \n",
" x = x % p # Update x if it is more \n",
" # than or equal to p \n",
" \n",
" while (y > 0): \n",
" \n",
" # If y is odd, multiply x with result \n",
" if (y & 1): \n",
" res = (res * x) % p \n",
" \n",
" # y must be even now \n",
" y = y >> 1 # y = y/2 \n",
" x = (x * x) % p \n",
" \n",
" return res \n",
" \n",
"# Utility function to store prime \n",
"# factors of a number \n",
"def findPrimefactors(s, n) : \n",
" \n",
" # Print the number of 2s that divide n \n",
" while (n % 2 == 0) : \n",
" s.add(2) \n",
" n = n // 2\n",
" \n",
" # n must be odd at this po. So we can \n",
" # skip one element (Note i = i +2) \n",
" for i in range(3, int(sqrt(n)), 2): \n",
" \n",
" # While i divides n, print i and divide n \n",
" while (n % i == 0) : \n",
" \n",
" s.add(i) \n",
" n = n // i \n",
" \n",
" # This condition is to handle the case \n",
" # when n is a prime number greater than 2 \n",
" if (n > 2) : \n",
" s.add(n) \n",
"\n",
"def findPrimitive( n) : \n",
" s = set() \n",
" \n",
" # Check if n is prime or not \n",
" if (isPrime(n) == False): \n",
" return -1\n",
" \n",
" # Find value of Euler Totient function \n",
" # of n. Since n is a prime number, the \n",
" # value of Euler Totient function is n-1 \n",
" # as there are n-1 relatively prime numbers. \n",
" phi = n - 1\n",
" \n",
" # Find prime factors of phi and store in a set \n",
" findPrimefactors(s, phi) \n",
" \n",
" # Check for every number from 2 to phi \n",
" for r in range(2, phi + 1): \n",
" \n",
" # Iterate through all prime factors of phi. \n",
" # and check if we found a power with value 1 \n",
" flag = False\n",
" for it in s: \n",
" \n",
" # Check if r^((phi)/primefactors) \n",
" # mod n is 1 or not \n",
" if (power(r, phi // it, n) == 1): \n",
" \n",
" flag = True\n",
" break\n",
" \n",
" # If there was no power with value 1. \n",
" if (flag == False): \n",
" return r \n",
" \n",
" # If no primitive root found \n",
" return -1\n",
"\n",
"n = 610292520201647\n",
"print(\"Smallest primitive root of\",n, \"is\", findPrimitive(n)) "
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Smallest primitive root of 610292520201647 is 5"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Code for Diffie–Hellman"
]
},
{
"cell_type": "code",
"execution_count": 1,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"g: 3 (a shared value), n: 610292520201647 (a prime number)\n",
"\n",
"Alice calculates:\n",
"a (Alice random): 69005\n",
"Alice value (A): 123944072679858 (g^a) mod p\n",
"\n",
"Bob calculates:\n",
"b (Bob random): 54992\n",
"Bob value (B): 191499579828593 (g^b) mod p\n",
"\n",
"Alice calculates:\n",
"Key: 312174094432853 (B^a) mod p\n",
"Key: 0368b0f4dd85126a039a6c7ba0fb35a56d12ec553b7f85df5c2914e907e71fb6\n",
"\n",
"Bob calculates:\n",
"Key: 312174094432853 (A^b) mod p\n",
"Key: 0368b0f4dd85126a039a6c7ba0fb35a56d12ec553b7f85df5c2914e907e71fb6\n"
]
}
],
"source": [
"import random\n",
"import hashlib\n",
"import sys\n",
"\n",
"g=3\n",
"p=610292520201647\n",
"\n",
"a=random.randint(10000,99999)\n",
"\n",
"b=random.randint(10000,99999)\n",
"\n",
"A = (g**a) % p\n",
"B = (g**b) % p\n",
"\n",
"print('g: ',g,' (a shared value), n: ',p, ' (a prime number)')\n",
"\n",
"print('\\nAlice calculates:')\n",
"print('a (Alice random): ',a)\n",
"print('Alice value (A): ',A,' (g^a) mod p')\n",
"\n",
"print('\\nBob calculates:')\n",
"print('b (Bob random): ',b)\n",
"print('Bob value (B): ',B,' (g^b) mod p')\n",
"\n",
"print('\\nAlice calculates:')\n",
"keyA=(B**a) % p\n",
"print('Key: ',keyA,' (B^a) mod p')\n",
"print('Key: ',hashlib.sha256(str(keyA).encode()).hexdigest())\n",
"\n",
"print('\\nBob calculates:')\n",
"keyB=(A**b) % p\n",
"print('Key: ',keyB,' (A^b) mod p')\n",
"print('Key: ',hashlib.sha256(str(keyB).encode()).hexdigest())"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Code to Find an integer k such that where a and m are relatively prime."
]
},
{
"cell_type": "code",
"execution_count": 17,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"2\n",
"1\n"
]
}
],
"source": [
"# Python3 program to calculate \n",
"# discrete logarithm \n",
"import math; \n",
"\n",
"# Iterative Function to calculate \n",
"# (x ^ y)%p in O(log y) \n",
"def powmod(x, y, p): \n",
"\n",
"\tres = 1; # Initialize result \n",
"\n",
"\tx = x % p; # Update x if it is more \n",
"\t\t\t# than or equal to p \n",
"\n",
"\twhile (y > 0): \n",
"\t\t\n",
"\t\t# If y is odd, multiply x with result \n",
"\t\tif (y & 1): \n",
"\t\t\tres = (res * x) % p; \n",
"\n",
"\t\t# y must be even now \n",
"\t\ty = y >> 1; # y = y/2 \n",
"\t\tx = (x * x) % p; \n",
"\treturn res; \n",
"\n",
"# Function to calculate k for given a, b, m \n",
"def discreteLogarithm(a, b, m): \n",
"\tn = int(math.sqrt(m) + 1); \n",
"\n",
"\tvalue = [0] * m; \n",
"\n",
"\t# Store all values of a^(n*i) of LHS \n",
"\tfor i in range(n, 0, -1): \n",
"\t\tvalue[ powmod (a, i * n, m) ] = i; \n",
"\n",
"\tfor j in range(n): \n",
"\t\t\n",
"\t\t# Calculate (a ^ j) * b and check \n",
"\t\t# for collision \n",
"\t\tcur = (powmod (a, j, m) * b) % m; \n",
"\n",
"\t\t# If collision occurs i.e., LHS = RHS \n",
"\t\tif (value[cur]): \n",
"\t\t\tans = value[cur] * n - j; \n",
"\t\t\t\n",
"\t\t\t# Check whether ans lies below m or not \n",
"\t\t\tif (ans < m): \n",
"\t\t\t\treturn ans; \n",
"\t\n",
"\treturn -1; \n",
"\n",
"# Driver code \n",
"a = 610292520201647; \n",
"b = 12542369; \n",
"m = 5; \n",
"print(discreteLogarithm(a, b, m)); \n",
"\n",
"a = 610292520201647; \n",
"b = 1852642; \n",
"m = 5; \n",
"print(discreteLogarithm(a, b, m)); \n",
"\n",
"# This code is contributed by mits \n"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Applying this code to see how long it takes to break your chosen 3 keys. Record your observation as shown in the table(min 3 (p,g) values and for each (p,g) 3 private keys (weak, medium strong and strong) "
]
},
{
"cell_type": "code",
"execution_count": 1,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"2\n",
"1\n"
]
}
],
"source": [
"# Python3 program to calculate \n",
"# discrete logarithm \n",
"import math; \n",
"\n",
"# Iterative Function to calculate \n",
"# (x ^ y)%p in O(log y) \n",
"def powmod(x, y, p): \n",
"\n",
"\tres = 1; # Initialize result \n",
"\n",
"\tx = x % p; # Update x if it is more \n",
"\t\t\t# than or equal to p \n",
"\n",
"\twhile (y > 0): \n",
"\t\t\n",
"\t\t# If y is odd, multiply x with result \n",
"\t\tif (y & 1): \n",
"\t\t\tres = (res * x) % p; \n",
"\n",
"\t\t# y must be even now \n",
"\t\ty = y >> 1; # y = y/2 \n",
"\t\tx = (x * x) % p; \n",
"\treturn res; \n",
"\n",
"# Function to calculate k for given a, b, m \n",
"def discreteLogarithm(a, b, m): \n",
"\tn = int(math.sqrt(m) + 1); \n",
"\n",
"\tvalue = [0] * m; \n",
"\n",
"\t# Store all values of a^(n*i) of LHS \n",
"\tfor i in range(n, 0, -1): \n",
"\t\tvalue[ powmod (a, i * n, m) ] = i; \n",
"\n",
"\tfor j in range(n): \n",
"\t\t\n",
"\t\t# Calculate (a ^ j) * b and check \n",
"\t\t# for collision \n",
"\t\tcur = (powmod (a, j, m) * b) % m; \n",
"\n",
"\t\t# If collision occurs i.e., LHS = RHS \n",
"\t\tif (value[cur]): \n",
"\t\t\tans = value[cur] * n - j; \n",
"\t\t\t\n",
"\t\t\t# Check whether ans lies below m or not \n",
"\t\t\tif (ans < m): \n",
"\t\t\t\treturn ans; \n",
"\t\n",
"\treturn -1; \n",
"\n",
"# Driver code \n",
"a = 610292520201647; \n",
"b = 12542369; \n",
"m = 5; \n",
"print(discreteLogarithm(a, b, m)); \n",
"\n",
"a = 610292520201647; \n",
"b = 1852642; \n",
"m = 5; \n",
"print(discreteLogarithm(a, b, m)); \n",
"\n",
"# This code is contributed by mits "
]
},
{
"cell_type": "code",
"execution_count": 4,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"54\n",
"49\n"
]
}
],
"source": [
"# Python3 program to calculate \n",
"# discrete logarithm \n",
"import math; \n",
"\n",
"# Iterative Function to calculate \n",
"# (x ^ y)%p in O(log y) \n",
"def powmod(x, y, p): \n",
"\n",
"\tres = 1; # Initialize result \n",
"\n",
"\tx = x % p; # Update x if it is more \n",
"\t\t\t# than or equal to p \n",
"\n",
"\twhile (y > 0): \n",
"\t\t\n",
"\t\t# If y is odd, multiply x with result \n",
"\t\tif (y & 1): \n",
"\t\t\tres = (res * x) % p; \n",
"\n",
"\t\t# y must be even now \n",
"\t\ty = y >> 1; # y = y/2 \n",
"\t\tx = (x * x) % p; \n",
"\treturn res; \n",
"\n",
"# Function to calculate k for given a, b, m \n",
"def discreteLogarithm(a, b, m): \n",
"\tn = int(math.sqrt(m) + 1); \n",
"\n",
"\tvalue = [0] * m; \n",
"\n",
"\t# Store all values of a^(n*i) of LHS \n",
"\tfor i in range(n, 0, -1): \n",
"\t\tvalue[ powmod (a, i * n, m) ] = i; \n",
"\n",
"\tfor j in range(n): \n",
"\t\t\n",
"\t\t# Calculate (a ^ j) * b and check \n",
"\t\t# for collision \n",
"\t\tcur = (powmod (a, j, m) * b) % m; \n",
"\n",
"\t\t# If collision occurs i.e., LHS = RHS \n",
"\t\tif (value[cur]): \n",
"\t\t\tans = value[cur] * n - j; \n",
"\t\t\t\n",
"\t\t\t# Check whether ans lies below m or not \n",
"\t\t\tif (ans < m): \n",
"\t\t\t\treturn ans; \n",
"\t\n",
"\treturn -1; \n",
"\n",
"# Driver code \n",
"a = 610292520201647; \n",
"b = 12542369; \n",
"m = 125; \n",
"print(discreteLogarithm(a, b, m)); \n",
"\n",
"a = 610292520201647; \n",
"b = 1852642; \n",
"m = 125; \n",
"print(discreteLogarithm(a, b, m)); \n",
"\n",
"# This code is contributed by mits "
]
},
{
"cell_type": "code",
"execution_count": 5,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"-1\n",
"-1\n"
]
}
],
"source": [
"# Python3 program to calculate \n",
"# discrete logarithm \n",
"import math; \n",
"\n",
"# Iterative Function to calculate \n",
"# (x ^ y)%p in O(log y) \n",
"def powmod(x, y, p): \n",
"\n",
"\tres = 1; # Initialize result \n",
"\n",
"\tx = x % p; # Update x if it is more \n",
"\t\t\t# than or equal to p \n",
"\n",
"\twhile (y > 0): \n",
"\t\t\n",
"\t\t# If y is odd, multiply x with result \n",
"\t\tif (y & 1): \n",
"\t\t\tres = (res * x) % p; \n",
"\n",
"\t\t# y must be even now \n",
"\t\ty = y >> 1; # y = y/2 \n",
"\t\tx = (x * x) % p; \n",
"\treturn res; \n",
"\n",
"# Function to calculate k for given a, b, m \n",
"def discreteLogarithm(a, b, m): \n",
"\tn = int(math.sqrt(m) + 1); \n",
"\n",
"\tvalue = [0] * m; \n",
"\n",
"\t# Store all values of a^(n*i) of LHS \n",
"\tfor i in range(n, 0, -1): \n",
"\t\tvalue[ powmod (a, i * n, m) ] = i; \n",
"\n",
"\tfor j in range(n): \n",
"\t\t\n",
"\t\t# Calculate (a ^ j) * b and check \n",
"\t\t# for collision \n",
"\t\tcur = (powmod (a, j, m) * b) % m; \n",
"\n",
"\t\t# If collision occurs i.e., LHS = RHS \n",
"\t\tif (value[cur]): \n",
"\t\t\tans = value[cur] * n - j; \n",
"\t\t\t\n",
"\t\t\t# Check whether ans lies below m or not \n",
"\t\t\tif (ans < m): \n",
"\t\t\t\treturn ans; \n",
"\t\n",
"\treturn -1; \n",
"\n",
"# Driver code \n",
"a = 610292520201647; \n",
"b = 12542369; \n",
"m = 63456456; \n",
"print(discreteLogarithm(a, b, m)); \n",
"\n",
"a = 610292520201647; \n",
"b = 1852642; \n",
"m = 465464; \n",
"print(discreteLogarithm(a, b, m)); \n",
"\n",
"# This code is contributed by mits "
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {},
"outputs": [
{
"ename": "MemoryError",
"evalue": "",
"output_type": "error",
"traceback": [
"\u001b[1;31m---------------------------------------------------------------------------\u001b[0m",
"\u001b[1;31mMemoryError\u001b[0m Traceback (most recent call last)",
"\u001b[1;32m<ipython-input-2-e09ae1952765>\u001b[0m in \u001b[0;36m<module>\u001b[1;34m\u001b[0m\n\u001b[0;32m 53\u001b[0m \u001b[0mb\u001b[0m \u001b[1;33m=\u001b[0m \u001b[1;36m12542369\u001b[0m\u001b[1;33m;\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m 54\u001b[0m \u001b[0mm\u001b[0m \u001b[1;33m=\u001b[0m \u001b[1;36m610292520201644\u001b[0m\u001b[1;33m;\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[1;32m---> 55\u001b[1;33m \u001b[0mprint\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mdiscreteLogarithm\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0ma\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mb\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mm\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m;\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m\u001b[0;32m 56\u001b[0m \u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m 57\u001b[0m \u001b[0ma\u001b[0m \u001b[1;33m=\u001b[0m \u001b[1;36m610292520201647\u001b[0m\u001b[1;33m;\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n",
"\u001b[1;32m<ipython-input-2-e09ae1952765>\u001b[0m in \u001b[0;36mdiscreteLogarithm\u001b[1;34m(a, b, m)\u001b[0m\n\u001b[0;32m 27\u001b[0m \u001b[0mn\u001b[0m \u001b[1;33m=\u001b[0m \u001b[0mint\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mmath\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0msqrt\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mm\u001b[0m\u001b[1;33m)\u001b[0m \u001b[1;33m+\u001b[0m \u001b[1;36m1\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m;\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m 28\u001b[0m \u001b[1;33m\u001b[0m\u001b[0m\n\u001b[1;32m---> 29\u001b[1;33m \u001b[0mvalue\u001b[0m \u001b[1;33m=\u001b[0m \u001b[1;33m[\u001b[0m\u001b[1;36m0\u001b[0m\u001b[1;33m]\u001b[0m \u001b[1;33m*\u001b[0m \u001b[0mm\u001b[0m\u001b[1;33m;\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m\u001b[0;32m 30\u001b[0m \u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m 31\u001b[0m \u001b[1;31m# Store all values of a^(n*i) of LHS\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n",
"\u001b[1;31mMemoryError\u001b[0m: "
]
}
],
"source": [
"# Python3 program to calculate \n",
"# discrete logarithm \n",
"import math; \n",
"\n",
"# Iterative Function to calculate \n",
"# (x ^ y)%p in O(log y) \n",
"def powmod(x, y, p): \n",
"\n",
"\tres = 1; # Initialize result \n",
"\n",
"\tx = x % p; # Update x if it is more \n",
"\t\t\t# than or equal to p \n",
"\n",
"\twhile (y > 0): \n",
"\t\t\n",
"\t\t# If y is odd, multiply x with result \n",
"\t\tif (y & 1): \n",
"\t\t\tres = (res * x) % p; \n",
"\n",
"\t\t# y must be even now \n",
"\t\ty = y >> 1; # y = y/2 \n",
"\t\tx = (x * x) % p; \n",
"\treturn res; \n",
"\n",
"# Function to calculate k for given a, b, m \n",
"def discreteLogarithm(a, b, m): \n",
"\tn = int(math.sqrt(m) + 1); \n",
"\n",
"\tvalue = [0] * m; \n",
"\n",
"\t# Store all values of a^(n*i) of LHS \n",
"\tfor i in range(n, 0, -1): \n",
"\t\tvalue[ powmod (a, i * n, m) ] = i; \n",
"\n",
"\tfor j in range(n): \n",
"\t\t\n",
"\t\t# Calculate (a ^ j) * b and check \n",
"\t\t# for collision \n",
"\t\tcur = (powmod (a, j, m) * b) % m; \n",
"\n",
"\t\t# If collision occurs i.e., LHS = RHS \n",
"\t\tif (value[cur]): \n",
"\t\t\tans = value[cur] * n - j; \n",
"\t\t\t\n",
"\t\t\t# Check whether ans lies below m or not \n",
"\t\t\tif (ans < m): \n",
"\t\t\t\treturn ans; \n",
"\t\n",
"\treturn -1; \n",
"\n",
"# Driver code \n",
"a = 610292520201647; \n",
"b = 12542369; \n",
"m = 610292520201644; \n",
"print(discreteLogarithm(a, b, m)); \n",
"\n",
"a = 610292520201647; \n",
"b = 1852642; \n",
"m = 610292520201644; \n",
"print(discreteLogarithm(a, b, m)); \n",
"\n",
"# This code is contributed by mits "
]
},
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